Faʻataʻitaʻiga o Fesili ma Talanoaga o Faʻatulagaga Mautū
O le integral mautinoa o se manatu autū i le calculus, e masani ona faʻaaogaina e suʻe ai le eria i lalo o se piʻo, fuafua le voluma o mea faigata, ma mo le tele o isi faʻaoga i inisinia ma fisiki. O le talanoaina o le integral mautinoa e le gata ina maua ai se malamalamaga faavae i lenei manatu ae faʻamalosia ai foʻi o tatou tomai i le auiliiliga faʻamatematika. O lenei tusiga e faʻamoemoe e tuʻuina atu faʻataʻitaʻiga o faʻafitauli o integral mautinoa faʻatasi ai ma talanoaga auiliili.
Manatu Faavae o le Fa'au'uga Mautinoa
A o le’i oo atu i fa’ata’ita’iga o fa’afitauli, se’i o tatou toe iloiloina nisi o manatu fa’avae o integrals mautinoa. O le integrals mautinoa, o lo’o fa’ailoa mai e le \(\int_a^bf(x) \, dx\), o lo’o fa’atusalia ai le eria i lalo o le pi’o o le galuega faatino \(f(x)\) mai le tulaga \(x = a\) i le tulaga \(x = b\).
I le fa'amatematika, o le integral mautinoa mai le \(a\) i le \(b\) o le galuega faatino \(f(x)\) e mafai ona fa'aalia e pei ona fa'aalia:
\[ \int_a^bf(x) \, dx = F(b) – F(a) \]
lea o le \(F(x)\) o le antiderivative o le \(f(x)\).
Fesili Fa'ata'ita'i ma Talanoaga
Se'i o tatou tilotilo i nisi o fa'ata'ita'iga o fa'afitauli o le integral mautinoa ma a latou talanoaga.
Fa'ata'ita'iga Fesili 1
Fesili:
Fuafua le integral mautinoa o le galuega faatino \(f(x) = 2x\) mai \(x = 1\) i le \(x = 3\).
Talanoaga:
Mo le foia o lenei integral, tatou te sailia muamua le antiderivative o le \(f(x) = 2x\).
O le antiderivative o le \(2x\) e faapea:
\[ F(x) = x^2 + C \]
Ae ui i lea, i totonu o integrals mautinoa tatou te le manaʻomia le tumau o le integration \(C\).
Ia, fa'aaoga tapula'a o integrals e fuafua ai:
\[ \int_1^3 2x \, dx = F(3) – F(1) \]
Fuafua le tau o le \(F(x)\) i luga o nei tapula'a:
\[ F(3) = 3^2 = 9 \]
\[ F(1) = 1^2 = 1 \]
O lea,
\[ \int_1^3 2x \, dx = 9 – 1 = 8 \]
Fa'ata'ita'iga Fesili 2
Fesili:
Fuafua le integral mautinoa o le galuega faatino \(f(x) = x^2 + 1\) mai \(x = 0\) i le \(x = 2\).
Talanoaga:
Saili le antiderivative o le \(f(x) = x^2 + 1\).
O le antiderivative o le \(x^2\) e faapea:
\[ \frac{1}{3}x^3 \]
O le antiderivative o le \(1\) o le \(x\).
O lea la, o le antiderivative o le \(f(x)\) e:
\[ F(x) = \frac{1}{3}x^3 + x \]
Ia, fa'aaoga tapula'a o integrals e fuafua ai:
\[ \int_0^2 (x^2 + 1) \, dx = F(2) – F(0) \]
Fuafua le tau o le \(F(x)\) i luga o nei tapula'a:
\[ F(2) = \frac{1}{3}(2)^3 + 2 = \frac{8}{3} + 2 = \frac{8}{3} + \frac{6}{3} = \frac{14}{3} \]
\[ F(0) = \frac{1}{3}(0)^3 + 0 = 0 \]
O lea,
\[ \int_0^2 (x^2 + 1) \, dx = \frac{14}{3} – 0 = \frac{14}{3} \]
Fa'ata'ita'iga Fesili 3
Fesili:
Fuafua le integral mautinoa o le galuega faatino \(f(x) = e^x\) mai \(x = 1\) i le \(x = 2\).
Talanoaga:
Saili le antiderivative o le \(f(x) = e^x\).
O le antiderivative o le \(e^x\) o le \(e^x\).
Ia, fa'aaoga tapula'a o integrals e fuafua ai:
\[ \int_1^2 e^x \, dx = F(2) – F(1) \]
Fuafua le tau o le \(F(x)\) i luga o nei tapula'a:
\[ F(2) = e^2 \]
\[ F(1) = e^1 = e \]
O lea,
\[ \int_1^2 e^x \, dx = e^2 – e \]
Fa'ata'ita'iga Fesili 4
Fesili:
Fuafua le integral mautinoa o le galuega faatino \(f(x) = \sin(x)\) mai \(x = 0\) i le \(x = \pi\).
Talanoaga:
Saili le antiderivative o le \(f(x) = \sin(x)\).
O le antiderivative o le \(\sin(x)\) o le \(-\cos(x)\).
Ia, fa'aaoga tapula'a o integrals e fuafua ai:
\[ \int_0^\pi \sin(x) \, dx = F(\pi) – F(0) \]
Fuafua le tau o le \(F(x)\) i luga o nei tapula'a:
\[ F(\pi) = -\cos(\pi) = -(-1) = 1 \]
\[ F(0) = -\cos(0) = -1 \]
O lea,
\[ \int_0^\pi \sin(x) \, dx = 1 – (-1) = 1 + 1 = 2 \]
Fa'ata'ita'iga Fesili 5
Fesili:
Fuafua le integral mautinoa o le galuega faatino \(f(x) = \frac{1}{x}\) mai \(x = 1\) i le \(x = e\).
Talanoaga:
Saili le antiderivative o le \(f(x) = \frac{1}{x}\).
O le antiderivative o le \(\frac{1}{x}\) o le \(\ln|x|\).
Ia, fa'aaoga tapula'a o integrals e fuafua ai:
\[ \int_1^e \frac{1}{x} \, dx = F(e) – F(1) \]
Fuafua le tau o le \(F(x)\) i luga o nei tapula'a:
\[ F(e) = \ln(e) = 1 \]
\[ F(1) = \ln(1) = 0 \]
O lea,
\[ \int_1^e \frac{1}{x} \, dx = 1 – 0 = 1 \]
I'uga
E ala i faʻataʻitaʻiga o loʻo i luga, ua matou faʻataʻitaʻiina le sailia o integrals mautinoa o galuega faʻavae eseese. I laʻasaga taʻitasi, e taua le muamua sailia o le antiderivative ona faʻaaoga lea o tapulaʻa o le integral e suʻe ai le tau mulimuli.
E taua tele le sao o integrals mautinoa i le tele o matā'upu o su'esu'ega ma fa'aoga fa'atino. O le malamalama i lenei manatu ma le fa'ata'ita'iina o fa'ata'ita'iga eseese o le a fa'amalosia ai ou tomai i le matematika.
